Great that you drove to fetch Mango rather than had him in cargo, especially as it does seem that he started out sick.
I've given the nasal vaccine before and not seen kittens develop runny noses in reaction. And one certainly wouldn't expect any such reaction to continue more than a day or so. Was this his first vaccination and how many days prior to pickup was he given this? And how old was he then?
I would imagine you would know if PCR testing was done, it's expensive. A lot of vets avoid it for that reason, it is my belief (especially with kittens) that it can be a false economy to avoid it...as sometimes then you treat with the wrong antibiotic. I'm glad you are also giving the Lysine but that is not a probiotic. Lysine is an amino acid that works to prevent growth of the herpes family of viruses. Rhinotracheitis (one of the things we vaccinate for) is in the Herpes family. So with the runny nose giving Lysine is a sensible thing to do. Make sure he's getting the right amount each day and not just as a "treat".
How long has Mango been taking the doxy? My concern is that because there was no testing this might not be the appropriate antibiotic. It may be that in your area, your vet has been seeing a lot of Mycoplasma infections so is starting with this, but given you drove a long way to get Mango it may not be the predominant pathogen where he was located. If you don't find he is responding, do take back to your vet to discuss this. It's not a general enough antibiotic to work for all respiratory infections. But on the other hand, for some respiratory infections where this is appropriate you might need to treat with the doxycycline for a month which is longer than most vets initially prescribe it for.
Yes you should have informed the breeder, for the reason you have mentioned (the other kittens) but also because the average contract has a 72 hour health guarantee that requires notifying the breeder.
No, just because your former cat has not had giardia does not mean there is none in your water. We ALL have different sensitivities to bugs, if my husband and I eat something, if anyone is to get food poisoning it will be me not him! Or he'll have a mild case and I will be laid out with it.
It is normal for people to fear the "unknown" and very common for them to associate "exotic cat" with "wild" and hence "feral behavior". You need to explain that the Serval is the most domesticatable of the SMALL exotic cats and that they don't see humans as prey but actually like hanging out with them...hence a Savannah is a interactive companion that just looks "wild" but behaves like a very energetic domestic cat. It's really important to stress that the Savannah is a domestic cat breed btw...there's a lot of Animal Rights Extremists (like HSUS and PETA) that love to play up the "wild" factor and try to get our breed banned.